The chapter "The Fokker-Planck Equation" by Shambhu N. Sharma and Hiren G. Patel discusses the Fokker-Planck equation, a key tool in describing the evolution of conditional probability density for Markov processes. The authors provide four different methods to derive the Fokker-Planck equation, including elementary proofs and proofs based on the Itô differential rule. They also discuss the application of the Fokker-Planck equation in various fields such as physics, mathematical control theory, and wireless communications. The chapter includes an analysis of a stochastic Duffing-van der Pol system, which is a second-order fluctuation equation describing a dynamical system in a noisy environment. The authors derive the Fokker-Planck equation for this system and present numerical simulations to validate their methods. The chapter concludes by highlighting the importance of the Fokker-Planck analysis in understanding and solving stochastic problems.The chapter "The Fokker-Planck Equation" by Shambhu N. Sharma and Hiren G. Patel discusses the Fokker-Planck equation, a key tool in describing the evolution of conditional probability density for Markov processes. The authors provide four different methods to derive the Fokker-Planck equation, including elementary proofs and proofs based on the Itô differential rule. They also discuss the application of the Fokker-Planck equation in various fields such as physics, mathematical control theory, and wireless communications. The chapter includes an analysis of a stochastic Duffing-van der Pol system, which is a second-order fluctuation equation describing a dynamical system in a noisy environment. The authors derive the Fokker-Planck equation for this system and present numerical simulations to validate their methods. The chapter concludes by highlighting the importance of the Fokker-Planck analysis in understanding and solving stochastic problems.